Rapid scattering simulation of objects in imaging using edge domain decomposition

ABSTRACT

A complex two-dimensional layout of a photomask or other three-dimensional object is systematically decomposed into a finite number of elementary two-dimensional objects with the ability to cause one-dimensional changes in light transmission properties. An algorithmic implementation of this can take the form of creation of a look-up table that stores all the scattering information of all two-dimensional objects needed for the synthesis of the electromagnetic scattered field from the original three-dimensional object. The domain is decomposed into edges, where pre-calculated electromagnetic field from the diffraction of isolated edges is recycled in the synthesis of the near diffracted field from arbitrary two-dimensional diffracting geometries. The invention has particular applicability in die-to-database inspection where an actual image of a mask is compared with a synthesized image that takes imaging artifacts of comers, edges and proximity into account. Another application is optical proximity correction which consists of evaluating the image of every feature on a mask and improving it by introducing edge shifts and iteratively adjusting the amounts of these shifts.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent applicationThis application is a Reissue Application of itsparent application Ser. No. 10/677,136, filed on Sep. 30, 2003, now U.S.Pat. No. 7,266,480 which claims priority from Provisional PatentApplication No. 60/415,510 filed Oct. 1, 2002 for “METHOD FOR RAPIDSCATTERING SIMULATION OF OBJECTS IN IMAGING VIA DOMAIN DECOMPOSITION ANDSPECTRAL MATCHING,” and is related to application Ser. No. 10/241,242filed Sep. 10, 2002 for “CHARACTERIZING ABERRATIONS IN AN IMAGING LENSAND APPLICATIONS TO VISUAL TESTING AND INTEGRATED CIRCUIT MASKANALYSIS,” all of which are incorporated herein by reference for allpurposes.

BACKGROUND OF THE INVENTION

This invention relates generally to optical imaging systems, and moreparticularly the invention relates to simulation and modeling ofelectromagnetic scattering of light in such imaging systems. Theinvention has applicability to optical proximity correction in photomasks and mask image inspection, but the invention is not limitedthereto.

The general components of an optical lithography tool, shownschematically in the diagram of FIG. 1 are the illumination system, theprojection system, the photomask (also called reticle), and thephotoresist, spun on top of a semiconductor wafer. The operationprinciple of the system is based on the ability of the resist to recordan image of the pattern to be printed. The mask, already carrying thispattern, is flooded with light and the projector forms an image of allmask patterns simultaneously onto (and into) the resist. The inherentparallelism of this process is the main reason why opticalphotolithography is favored over any other lithography, since itfacilitates a very high throughput of 30-120 wafers per hour. The lightintensity distribution on top of the resist surface is commonly referredto as aerial image. The resist itself is a photosensitive material whosechemical composition changes during light exposure. The pattern isthereby stored in form of a latent (bulk) image within the resist. Afterexposure has occurred, the resist is developed by means of a chemicalprocess that resembles the process of developing photographic film.After development, the exposed parts of the resist remain or dissolvedepending on its polarity (negative or positive, respectively). Theend-result of the lithography process is a more or less exact (scaled ornot) replica of the mask pattern on the wafer surface that will play therole of a local protective layer (mask) for subsequent processing steps(etching, deposition, implantation).

The role of the illumination system is to deliver a light beam thatuniformly trans-illuminates the entire reticle. See FIG. 2(a). Ittypically consists of various optical elements, such as lenses,apertures, filters and mirrors. The light source is responsible forgenerating very powerful and monochromatic radiation. Power is necessarybecause it is directly related to throughput. Monochromaticity isimportant because high quality refractive (or reflective, in the case ofEUV lithography) optics can only be fabricated for a very narrowillumination bandwidth. State-of-the-art optical lithography toolsemploy excimer lasers as their light source. Deep Ultra Violet (DUV)lithography is the term used for lithography systems with illuminationwavelengths λ=248 nm (excimer laser with KrF), λ=193 nm (excimer laserwith ArF) and λ=157 nm (excimer laser with F₂). The successfuldevelopment of current and future optical photolithography technologiesis hinged upon research advances in both excimer laser technology andnovel materials that possess the required properties (high opticaltransmission at DUV wavelengths, thermal properties, stability afterheavy DUV radiation exposure) by which the optical elements of thesystem will be made.

All illumination systems in optical projection printing tools aredesigned to provide what is known as Köhler illumination. By placing thesource or an image of the source in the front focal plane of thecondenser column, the rays originating from each source point illuminatethe mask as a parallel beam, as seen in FIG. 2. Here, each source pointemits a spherical wave that is converted by the illumination system intoa plane wave incident on the object (photomask). The angle of incidenceof the plane wave depends on the location of the source point (α,β) withrespect to the optical axis (0,0). Each parallel beam is a plane wavewhose direction of propagation depends on the relative position of thesource point with respect to the optical axis. Nonuniformity in thebrightness of the source points is averaged out so that every locationon the reticle receives the same amount of illumination energy. As wewill see in subsequent Sections Köhler illumination can be modeled in aconcise mathematical way.

In addition to dose uniformity, the lithography process should alsomaintain directional uniformity such that the same features arereplicated identically regardless of their orientations. The shape ofthe light source is therefore circular (or rotationally symmetric) intraditional optical lithography, although this is not true for certainadvanced illumination schemes such as quadrupole illumination, wheredirectional uniformity is sacrificed in order to maximize the resolutionof features with certain orientations.

The coherence of the light source is another important attribute.Temporal coherence is usually not a big concern, since the narrowbandwidth of excimer lasers implies high temporal coherence. Spatialcoherence (or just plain coherence) on the other hand is alwayscarefully engineered and, in most cases, adjustable. Using specialscrambling techniques, the light emitted from any point of the source ismade completely uncorrelated (incoherent) to the light emitted fromevery other point. However, light gathers coherence as it propagatesaway from its source. The frequently quoted partial coherence factor ais a characteristic of the illumination system and is a measure of thephysical extent and shape of the light source. The larger the lightsource, the larger the partial coherence factor, and the light sourcehas a lower degree of coherence. In the limit of an infinite source,imaging is incoherent and σ=∞. On the other hand, the smaller the lightsource, the smaller the partial coherence factor, and the higher thedegree of coherence. Imaging with a point source is fully coherent andσ=0. Note that a point source in a Köhler illumination will result in asingle plane wave illuminating the mask and the angle of incidence ofthis wave depends on the relative position of the point source withrespect to the optical axis. For partial coherence factors between zeroand infinity, imaging is partially coherent. Typical partial coherencefactors in optical lithography range from 0.3 to 0.9.

The projection system typically consists of a multi-element lens column(up to 30-40 lenses) that may also have apertures, filters, or otheroptical elements, and it is a marvel of engineering precision in orderto be able to reliably project images with minimum dimensions on theorder of 100 nm for state-of-the-art systems. One of the main reasonsfor the required high precision is control of the aberrations ordeviations of the wavefront from its ideal shape. Two relevantparameters of the projection system are the numerical aperture, NA, andthe reduction factor, R. The numerical aperture is, by definition, thesine of the half-angle of the acceptance cone of light-rays as seen fromthe image side of the system. The ratio of image-height to object-heightis, by definition, the magnification factor M of the system. The inverseof the magnification factor is the reduction factor, R. Since a typicalsystem in photolithography projects at the image plane a scaled-downversion of the object (mask), M is less than 1 and R is greater than 1.State-of-the-art systems currently have reduction factors of R=4 or 5.Note that two numerical apertures exist in the projection system, namelyNA_(i) (or simply NA) and NA_(o), which refer to the half-angle of theacceptance cone as seen from the image side and from the object (mask)side, respectively. They are related through the reduction factor asfollows:

$\begin{matrix}{R = {\frac{{NA}_{i}}{{NA}_{o}}.}} & {{Equation}\mspace{14mu}\text{2-1}}\end{matrix}$

For a circularly shaped light source, the partial coherence factor σmentioned above is related to the numerical apertures of both theprojection system and the illumination system. Specifically, σ is givenby:

$\begin{matrix}{\sigma = {\frac{{NA}_{c}}{{NA}_{p}}.}} & {{Equation}\mspace{14mu}\text{2-2}}\end{matrix}$where NA_(c) is the numerical aperture of the condenser lens(illumination system) and NA_(p) is the numerical aperture of theprojector lens. Some confusion arises from the fact that, in the aboveequation, the reduction factor of the imaging system is implicitly takeninto account. FIG. 2(b) clarifies the situation by showing simplifieddiagrams of two optical systems with parameters NA=0.5, σ=0.5 and R=5 orR=1.

The photomask, also called reticle, carries the pattern to be printed ata given lithography processing step. The masks of integrated circuitshaving large die-sizes or footprints (that is, occupying large areas onthe semiconductor wafer), typically carry just one copy of the chippattern. A matrix of several chip patterns is contained in one maskwhenever the chip size permits. Note that the mask is drawn R times theactual size on the semiconductor wafer, since the dimensions of thecircuit will be scaled down by the reduction factor, R. For this reasonit is not sufficient to just provide feature sizes, since it may not beimmediately obvious from the context whether these are photomask(object) or resist (image) sizes. A typical convention fordistinguishing photomask feature sizes from resist feature sizes is toinclude in parenthesis the reduction factor R. For example, a 600 nm(4×) line has a size of 600 nm on the mask, and would produce a 600nm/4=150 nm line if used in a 4×imaging system. Similarly, a 130 nm (1×)line refers to the size of a line at the image (wafer) plane and wouldresult from the printing of a 130 nm line on the mask for a system withR=1, or a 520 nm (4×130 nm) line on the mask for a system with R=4, or a1.3 μm (10×130 nm) line on the mask for a system with R=10.

Depending on their operation principle, photomasks can be divided intotwo broad categories: conventional binary or chrome-on-glass (COG) masksand advanced phase-shifting masks (PSM).

A binary or COG mask consists of a transparent substrate (mask blank),covered with a thin opaque film that bears the desired pattern. Lightcan either pass unobstructed through an area not covered by the opaquefilm or be completely blocked if it is incident on an area that isprotected by the film. This binary behavior of the transmissioncharacteristic of the mask is responsible for its name. The mask blankfor DUV lithography typically consists of fused silica glass that hasexcellent transmission at λ=248 nm and somewhat poorer but acceptabletransmission at λ=193 nm and λ=157 nm. The opaque film is typically onthe order of 100 nm thick and has a chromium (Cr) composition.

Adding phase modulation to the photomask can profoundly increase theattainable resolution. This is the principle followed by phase-shiftingmasks (PSM), which employ discrete transmission and discrete phasemodulation. There are many different flavors of PSMs depending on theway that the phase modulation is achieved. One of the most promising PSMtechnologies is what is known as alternating phase-shifting mask (alt.PSM, or APSM). Here are cut-planes of geometry of a binary (COG) mask(a) and an alternating phase-shift mask (b). The ideal electric fielddistribution for the binary mask (c) leads to a poor image intensitydistribution (e) at the image plane, whereas the ideal electric fielddistribution for the alt. PSM (d), because of destructive interference,leads to a robust image. The principle of an alt. PSM is compared withthat of a binary mask in FIG. 3. The center line is bordered bytransmitting regions with 180° phase difference on an alt. PSM and byclear areas of the same phase on a binary mask. The phase difference onthe alt. PSM leads to destructive interference, resulting in a sharpdark image. The binary mask image is not as sharp because of the lack ofphase interaction. The 180° phase difference is created by etchingtrenches, also called phase-wells, into the fused silica substrateduring the alt. PSM fabrication process, which is now more complex thanthe COG fabrication process. The difference in the amount of materialremoved d_(etch) is such that the path length difference between lightpassing through the different phase regions is half of the wavelength inair.

Sub-wavelength lithography, where the size of printed features issmaller than the exposure wavelength, places a tremendous burden on thelithographic process. Distortions of the intended images inevitablyarise, primarily because of the nonlinearities of the imaging processand the nonlinear response of the photoresist. Two of the most prominenttypes of distortions are the wide variation in the linewidths ofidentically drawn features in dense and isolated environments (dense-isobias) and the line-end pullback or line-end shortening (LES) from drawnpositions. The former type of distortion can cause variations in circuittiming and yield, whereas the latter can lead to poor current tolerancesand higher probabilities of electrical failure.

Optical proximity correction or optical proximity compensation (OPC) isthe technology used to compensate for these types of distortions. OPC isloosely defined as the procedure of compensating (pre-distorting) themask layout of the critical IC layers for the lithographic processdistortions to follow. This is done with specialized OPC software. Inthe heart of the OPC software is a mathematical description of theprocess distortions. This description can either be in the form ofsimple shape manipulation rules, in which case the OPC is referred to as“rule-based OPC,” or a more detailed and intricate process model for a“model-based OPC.” The OPC software automatically changes the masklayout by moving segments of line edges and adding extra features that(pre-) compensate the layout for the distortions to come. Although afterOPC has been performed the mask layout may be quite different than theoriginal (before OPC) mask, the net result of this procedure is aprinted pattern on the wafer that is closest to the IC designer'soriginal intent.

In the early 1990's the problem of OPC for large mask layouts wasaddressed formally as an optimization problem. The size of such aproblem is formidable, but through introduction of appropriateconstraints—induced primarily from mask fabrication constraints, localoptimal mask “points” were successfully demonstrated. This research hasled to commercially available software tools that perform OPC on afull-chip scale. All three approaches rely heavily on speedycalculations of the image intensity at selected points of the imagefield. Although the methods by which they achieve this appear seeminglydifferent, they are nevertheless the same, in the sense that they allrely on a decomposition of the kernel of the imaging equation forpartially coherent light.

Domain decomposition techniques, where a large electromagnetic problemis broken up into smaller pieces and the final solution is arrived at bysynthesizing (field-stitching) the elemental solutions, have beenproposed for the study of one-dimensional binary (phase only)diffractive optical elements. Others working on the same problem havedemonstrated how to create and use a perturbation model for binaryedge-transitions based on the product of the ideal, sharp transition andthe continuous field variations in the vicinity of the edge.Independently, a similar technique to field-stitching has been proposedfor the simulation of two-dimensional layouts of advanced photomasks(alternating PSM, masks with OPC) that properly models interactions fromneighboring apertures and furthermore takes advantage of the spectralproperties of the diffracted fields to come up with a compact model forthe edge-transitions.

The present invention is directed to an improved methodology for OPC onoptical masks.

SUMMARY OF THE INVENTION

In accordance with the invention, a complex two-dimensional layout of aphotomask or other three-dimensional object is systematically decomposedinto a finite number of elementary two-dimensional objects with theability to cause one-dimensional changes in light transmissionproperties. An algorithmic implementation of this can take the form ofcreation of a look-up table that stores all the scattering informationof all two-dimensional objects needed for the synthesis of theelectromagnetic scattered field from the original three-dimensionalobject.

More particularly, the domain is decomposed into edges, wherepre-calculated electromagnetic field from the diffraction of isolatededges is recycled in the synthesis of the near diffracted field fromarbitrary two-dimensional diffracting geometries. In this invention, thesum of the complex fields is taken rather than the product due to thelinearity of the Kirchoff-Fresnel diffraction integral, which makes thedecomposition method possible.

The invention has particular applicability in die-to-database inspectionwhere an actual image of a mask is compared with a synthesized imagethat takes imaging artifacts of comers, edges and proximity intoaccount. A very important current challenge in synthesizing images isthe correct evaluation of electromagnetic effects at edges ofphase-shifted mask features. These electromagnetic effects are a strongfunction of the edge topography of the nonplanar features introducedduring the mask writing process. The inspection of every feature on a50M transistor chip must be accomplished in an inspection time on theorder of 5 hours for feature sizes of 100 nm. Future inspection ratesare expected to hit 200M inspection pixels per hour, which would allow amask with 50 nm features and 200M transistors to be inspected at a pixelsize of 25 nm in about 5 hours.

Another application is optical proximity correction which consists ofevaluating the image of every feature on a mask and improving it byintroducing edge shifts and iteratively adjusting the amounts of theseshifts. A typical OPC correction process on a 50M transistor mask takesabout a full day. Since the number of iterations in the OPC shifts is onthe order of 5 the full mask image is computed on average roughly at thesame speed of the die-to-database inspection, or 50M transistors every 5hours for the 100 mn generation and 200M transistors per hour for the 50nm generation.

The invention and objects, features and applications thereof will bemore apparent from the following detailed description and appendedclaims when taken with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general diagram of an optical photolithography tool.

FIG. 2(a) illustrates the illumination system of FIG. 1 with Köhler'smethod.

FIG. 2(b) illustrates numerical apertures of illumination and projectionlenses in the tool of FIG. 1.

FIGS. 3(a)-3(f) are a comparison of operational principles of analternating phase shift mask (alt. PSM) and a binary (COG) mask as usedin the invention.

FIG. 4 illustrates light decomposition of an opening into two edges.

FIGS. 5(a)-5(f) illustrate an edge domain decomposition method inaccordance with the invention.

FIGS. 6(a)-6(f) illustrate an edge domain decomposition method appliedon an isolated line.

FIG. 7 illustrates an edge domain decomposition method applied on anarbitrary polygon.

FIGS. 8(a)-8(d) illustrate dependence of edge scattering on edge profileand polarization.

FIGS. 9(a)-9(d) illustrate complex differences of true edge-scatteringfrom ideal and sharp transition.

FIGS. 10(a)-10(d) illustrate the spectra of complex differences shown inFIGS. 9A-9D.

FIG. 11 illustrates edge domain decomposition on a large mask layout.

FIGS. 12(a)-12(c) illustrate edge domain decomposition applied on anisolated square hole (scattered field).

FIGS. 13(a)-13(c) illustrate edge domain decomposition applied on anisolated square hole (spectrum).

FIGS. 14(a)-14(c) illustrate edge domain decomposition applied on anisolated square island (scattered field).

FIGS. 15(a)-15(c) illustrate edge domain decomposition applied on anisolated square island (spectrum).

FIG. 16 illustrates edge domain decomposition on a large mask layout(images).

FIGS. 17(a), 17(b) illustrate matched bandwidth approximation to edgediffraction.

FIGS. 18(a), 18(b) illustrate matched bandwidth approximation applied tothe hole of FIG. 12.

FIGS. 19(a), 19(b) illustrate defect assessment using prior arttechniques.

FIG. 20 illustrates domain decomposition in accordance with anembodiment of the invention when a defect is present.

FIGS. 21(a), 21(b) illustrate layout and amplitude of scattered fieldbelow the mask at the observation plane.

FIGS. 22(a)-22(d) illustrate a 90 degree phase defect present in thecenter of the bottom-left hole of a mask.

FIGS. 23(a)-23(d) illustrate a defect present in the center of thebottom-right hole of a mask.

FIGS. 24(a)-24(b) are a comparison of the decomposition method with therigorous determined scattering.

FIGS. 25(a)-25(b) are plan views of defective masks of FIGS. 22 and 23.

FIGS. 26(a)-26(b) are spectra plots (magnitude) of errors of FIGS. 22and 23.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Adam and Neureuther, “SIMPLIFIED MODELS FOR EDGE TRANSITIONS IN RIGOROUSMASK MODELING,” 26^(th) Annual International Symposium onMicrolithography, Mar. 1, 2001, discloses domain decomposition methods(DDM) that allow the division of a larger mask diffraction problem intoa set of constituent parts or single openings in the mask.

The present invention provides an extension of the disclosed domaindecomposition methods (DDM). A domain decomposition method based onedges uses pre-calculated electromagnetic field from the diffraction ofisolated edges which is recycled in the synthesis of the near diffractedfield from arbitrary two-dimensional diffracting geometries. Although atfirst glance it might be tempting to discard edge-decompositiontechniques as inaccurate for masks with large vertical topography andfeatures with small lateral dimensions, it will be shown that for alarge set of practical situations, they are accurate. A key differencein this method from the prior art is that the sum of the complex fieldsis taken instead of the product. The reason for this is in the linearityof the Kirchhoff-Fresnel diffraction integral, and this linearity makespossible the decomposition method.

A natural extension of the DDM that is also based on the linearity ofthe Kirchhoff-Fresnel diffraction integral is shown in FIG. 4. Anopening with transmission T and phase φ in an infinitely thin, opaquescreen can be decomposed by way of the linearity of theKirchhoff-Fresnel diffraction integral into two edges. The uniform planewave illumination is subtracted to restore the light level everywhere.The 1D mask layout is first decomposed into single opening masks, asnoted above, and subsequently each opening is decomposed into two edgesand a uniform illumination field as in FIG. 4. Such a decomposition canbe proven to be exact for infinitely thin and perfectly conductingscreens through the use of the electromagnetic form of Babinet'sprinciple, but it should break down if applied to photomasks with“thick” vertical structures (compared to λ) and not perfectly absorbingscreen materials. Also, because of the reduced ability of light topenetrate through small (compared to λ) openings, it is expected tobreak down for small mask features. This decomposition method will behereafter referred to as the edge domain decomposition method oredge-DDM. The application of the edge-DDM to a single isolated space andan isolated line are described next, followed by a systematic evaluationof the limits of applicability of the method.

Consider the isolated mask space (opening) shown in FIGS. 5(a)-5(f). Theamplitude of the complex Ey-field under Ey (TE) illumination is shown inFIG. 5(c) and the scattered field across the observation plane is shownin FIG. 5E. According to FIG. 4, the field across the observation planecan be alternatively obtained by using the scattered field from an edgethat has the same vertical profile as the space, as shown in FIGS. 5(b)and 5(d). The resulting field from this edge-DDM is overlaid on the plotof FIG. 5E along with the field from the edge-scattering. The normalizedmean square error 1 (NMSE) between the r-mask and the edge-DDM is 0.19%.In FIG. 5(f) the spectra of the r-mask and edge-DDM are compared and itis seen that they differ by 0.13% in a NMSE sense if the whole spectrumof propagating waves is considered and by merely 0.007% within thecollection ability of a projection system with NA=0.8 and R=4. Sinceonly the portion of the spectrum that is collected by the projectionsystem contributes to the image formation, the exceptional accuracy ofthe edge-DDM is this regime is more than adequate for accurate imagesimulations.

Similar steps can be followed for the isolated line shown in FIG. 6(a)illuminated with Ex (TM) polarized light. Note that it is not typical tohave a line as shown here, where in both sides the glass has beenetched. Normally only one side of the line is phaseshifted, but thisexample simply aims to introduce the principle of the edge-DDM. A linewith one side phase-shifted and the other left intact can still bedecomposed via the edge-DDM using the diffraction fields from the twodifferent edges. The amplitude of the complex Ex field everywhere in thedomain of FIG. 6(a) is shown in FIG. 6(c) and for the edge of FIG. 6(b)in FIG. 6(d) respectively. The true (r-mask) scattered field is comparedwith the synthesized field with edge-DDM in FIG. 6(e) and the spectraare compared in FIG. 6(f). The error (NMSE) is seen to be small in boththe near field and spectra plots and specifically the error of thethrough-the-lens (TTL) spectrum is just 1.5E-5.

In the examples of FIG. 5 and FIG. 6 the edge-DDM is seen to be accuratecompared with the simulation of the exact mask structure. Although thevertical mask topography (180°≈170 nm≈1λ) is comparable to thewavelength, the fact that the lateral dimensions of the space and theline are relatively large compared with the wavelength (CD=400 nm>2×193nm=2×λ) renders the edge-DDM accurate. One expects that as the lateralsizes become smaller and the vertical mask topography even larger theedge-DDM will eventually break down. The limits of the edge-DDM wereinvestigated through exhaustive simulation of isolated spaces and lineson the mask with various etched depths and amounts of undercut. Thesidewall angles were always kept vertical and no rounding at the bottomof the etched wells was included. The mask CD was varied in the set {2μm, 1 μm, 0.6 μm, 0.4 μm, 0.2 μm(˜1λ), 0.1 μm}, the etched depth in {0°,90°, 180°, 270°, 360°} and the amount of underetch in {0 nm, 25 nm, 50nm, 75 nm, 100 nm}. All 2 (space/line) by 2 (TE/TM illumination) by 6(CD sizes) by 5 (etched depths) by 5 (underetch)=600 cases were run andthe rigorous (r-mask) scattered field was automatically compared withthat obtained from edge-DDM. The results of the investigation werespectacular. The edge-DDM can readily achieve better than 1% accuracy(in a NMSE sense) in the near field for CD=400 nm (4×) or largerregardless of depth, underetch and polarization. The accuracy in theTTL-spectrum is sufficient in those cases even for inspection (R=1)simulations. For lines, the edge-DDM maintains excellent accuracy in theTTL-spectrum with R=4 down to a mask CD of 100 nm (˜0.5λ) with 90° ofetched depth and 0 nm of undercut, or mask CD of 200 nm (˜1λ) with 270°of etched depth and 25 nm of undercut. For spaces, the accuracy of themethod is acceptable at least down to mask CD of 200 nm (˜1λ) with 270°of etched depth and 50 nm of underetch.

Current state-of-the-art photomasks and processes utilize sub-resolutionassist features that are never less than one wavelength in size on themask. This clearly implies that the edge-DDM has the potential toaccurately simulate the most advanced photomask technologies (OPC, alt.PSM, OPC on alt. PSM). Again, the true power of the method lies on itsapplication in 2D mask layouts. However, one key difference with the DDMof mask openings is the following: only one rigorous 2D simulation of anedge suffices for the scattered field reconstruction of any arbitrarysize opening on the mask. The DDM of openings requires a separatesimulation of every different mask size. Next, the application of theedge-DDM in 2D layouts is developed.

The application of the edge-DDM in 2D layouts is straightforward. A masklayout comprises of a large number of edges positioned at differentlocations and having different orientations. No matter how complicatedthe mask technology, there are usually only a small number of differenttypes of edges present in the layout. For example, a single exposure0°/90°/270° alt. PSM has five types of edges: i) Cr-layer/0° edge, ii)Cr-layer/90° edge, iii) Cr-layer/270° edge, iv) 0°/90° edge and 0°/270°edge. Depending on the orientation of each edge in the layout it “sees”and responds to the incident field differently. If the illuminatingfield is a TE (E_(y)) normally incident plane wave and the edge isoriented along the y-axis it “sees” TE illumination, but if it isoriented along the x-axis it “sees” TM illumination. This is illustratedin FIG. 7. Depending upon the orientation of each edge in the layout it“sees” and responds to the incident field differently. Here theilluminating field is a TE (E_(y)) normally incident plane wave and theedges that are oriented along the y-axis “see” TE illumination (parallelto the edge), whereas the edges that are oriented along the x-axis “see”TM illumination (perpendicular to the edge). A separate 2D edgesimulation is required to capture the different response of the edge todifferent field polarizations. The response of each edge to itsrespective illumination is taken into account rigorously from apre-stored 2D edge-diffraction simulation. However, corner effects areeffectively ignored, since the finite extent of each edge is notrigorously taken into account. Instead, the scattered field at the endpoints of every edge is abruptly terminated (truncated) in a perfectsquare-wave fashion to the field value of the kmask model. Comer effectsin typical imaging situations are mapped at the extremities of thespectra and do not contribute to the image formation. Owing to thisobservation, the edge-DDM applied in 2D layouts is successful inaccurately capturing the true electromagnetic behavior of edges in arapid manner, while the insignificant comer effects are safely ignored.

Slightly different embodiments of the edge-DDM applied in 2D layouts canbe devised. One that is powerful and general in its application isdescribed next. Then, scattering results from various edge profiles thatwould be encountered in alt. PSMs are presented for completeness. Next,application of the edge-DDM in simple 2D mask layouts is presented andthe results of the method are compared with fully rigorous r-mask modelsin order to establish the validity of this approach. Finally, a largeportion of a real layout from a single exposure 0°/90°/270° alt. PSMthat is too big for 3D rigorous simulation is simulated via the edge-DDMand the aerial image is compared with the simple k-mask model.

First, every distinct edge that is present on the layout ispre-simulated with all possible illumination directions (fieldpolarizations) that are required, based on the orientations of the edgeencountered within the layout. For simplicity only Manhattan-typelayouts are considered, where the edges are oriented along either the x-or the y-axis and require electromagnetic simulation under both paralleland perpendicular to the edge polarizations. The extension tonon-Manhattan layouts although much more computationally laborious isstraightforward. Subsequently, each distinct mask layer is broken upinto a set of mutually disjoint (non-overlapping) rectangles. Forexample, a plain binary mask consists of two layers, namely theabsorption layer (typically Cr-based) and the clear (uncovered) layer,whereas a more advanced 0°/90°/270° three-phase alt. PSM consists offour layers, namely the absorption layer and the 0° clear, 90° clear and270° clear layers. Next, looping through all rectangles of each layer,the type of each one of the four edges of the rectangle is determinedbased on what its neighboring layer is. Trivial edges (when theneighboring rectangle is of the same type) are discarded. It is alsopossible that only part of a single edge from a rectangle is neighboringwith one layer and the rest is neighboring with a different layer (ordifferent layers). In that case the edge is broken up in multiple edges,in a way that tracks the neighboring layer. What is left at the end ofthis process is a set of all non-trivial edges that are present in thelayout (location, size and orientation) and their type, i.e. which arethe two mask layers on each side of the edge. Given the mask layout, thek-mask (Kirchhoff-mask) model is readily available. Finally, therespective (complex) difference of the true edge-scattering from theideal and sharp k-mask model is added to all non-trivial edgesaccordingly. This revised mask model that results from the edge-DDMincludes accurate information about the electromagnetic scattering fromthe edges and it is a quasi-rigorous mask model (qr-mask) in the samesense that the term was used where only 2D mask simulations were used toapproximate a 3D electromagnetic problem.

The above algorithm was implemented in a MATLAB environment. The partthat deals with the polygon and edge extraction can probably be judgedas rudimentary and most likely cannot compete in speed with moresophisticated implementations in production caliber CAD software, but itsuffices for the purposes of this work, that is, to demonstrate theproof of concept of the edge-DDM and the feasibility of includingaccurate edgescattering information in rapid aerial image simulations.

Examining the departure of the scattered field from an edge from itsideal, sharp edge-transition and how it depends on the profile of theedge and the incident field polarization can provide intuition of thephysical mechanisms involved.

FIG. 8 shows the amplitude of the (near) scattered field from an edgewith various profiles. The ideal, sharp edge-transition is also shown onevery graph for comparison. In FIG. 8(a) and (b) the edge-scattering fordifferent etched depths (0°, 180° and 360°) and vertical sidewalls withno undercut is shown for TE (E_(y)) and TM (E_(x)) illuminationrespectively. The effect of undercut is depicted in FIG. 8(c) and (d),where the edgescattering for different amounts of undercut (0 nm, 25 nm,50 nm and 75 nm) and vertical sidewalls with 180° etched depth is shownfor TE and TM illumination respectively. It is interesting to note thatincreasing the etched depth is “making the edge response slower”,meaning that the near scattered field from the edge requires moredistance to reach its “on” value from its “off” value, although thiseffect is more pronounced for the first increment from 0° to 180° ratherthan from 180° to 360°. Similarly, increasing the amount of undercutappears to “make the edge response faster” and furthermore this behaviorappears to be linear, since for every increment of the undercut theedge-transition gains a constant distance in its “off-on” behavior.

It is instructive to isolate the difference of the true edge-scatteringfrom the ideal, sharp edge-transition. This is done in FIG. 9, whichdepicts the amplitude of this (complex) difference for all respectiveplots of FIG. 8. Here the amplitude of the complex difference of thetrue electromagnetic field scattered from an edge from the ideal, sharptransition (step-function) is shown in FIGS. 9(a)-(d) for all edgeprofiles and illumination polarizations of FIG. 8 respectively. FIGS.9(a) and (c) are for TE polarization and FIGS. 9(b) and (d) are for TMpolarization. The spatial extent of the difference grows with increasingetched depths, as seen in FIGS. 9(a) and (b), and shrinks withincreasing undercuts, as seen in FIG. 9(d) for TM polarization. Thebehavior in FIG. 9(c) appears somewhat erratic, but if the phase is alsotaken into account, it is shown in FIG. 10 that this is also the casefor TE polarization. Observe that, indeed, by increasing the etcheddepth the difference has a larger spatial extent, whereas increasing theamount of undercut confines the spatial extent of the difference. Thisis crucial, because it is exactly the spatial extent of the differencethat affects its spectral distribution. If the difference is wellconfined, its spectral distribution is spread out to higher spatialfrequencies and the lower frequency content is small. In such a case thecomplex difference can be neglected and the approximation of the ideal,sharp transition is adequate.

However, when the difference has a larger spatial extent, then its lowerfrequency content becomes significant and it cannot be neglected. Thisis shown in FIG. 10, which depicts the magnitude of the spectra of allrespective difference distributions of FIG. 9. Here the spectra of thecomplex differences shown in FIG. 9 are depicted here respectively.FIGS. 10(a) and (c) are for TE polarization and FIGS. 10(b) and (d) arefor TM polarization. Observe that the through-the-lens (TTL) parts ofthese spectra, which for typical lithographic imaging situations (R−4,NA˜0.7-0.8) extent up to ˜±0.008 (nm)⁻¹ are smaller for smaller etcheddepths and larger undercuts, regardless of polarization, as expectedfrom the observations on the plots of FIG. 8 and FIG. 9. Observe thatthe TTL spectrum of the difference attains larger values when the etcheddepth increases and smaller values when the amount of undercutincreases, as expected from FIG. 8 and FIG. 9. Remember that thenormalization used in spectra plots is that a constant (DC) value of 1(clear field) would produce a Dirac δ-function at zero frequency withamplitude of 19.4 (√{square root over (η)}=√{square root over(√μ₀/ε₀)}≈19.4 Ω).

The application of the edge-DDM on an isolated square hole and anisolated square island (post) and comparison of the results with fullyrigorous 3D mask simulations (rmasks) are presented here. The edgeprofile has a glass etched depth of 180° and an undercut of 50 nm. Thedimensions of both the hole and the island are 400 nm×400 nm (4×),corresponding to approximately 2×2 wavelengths, for λ=193 nm.

FIG. 12(a) and (b) depict the amplitude of the scattered field acrossthe observation plane below the hole obtained via the edge-DDM and viafully rigorous 3D simulation, respectively. The amplitude of theirdifference (error of the edge-DDM) is shown in FIG. 12(c). Locally theamplitude of the error is seen to reach a discouraging level of almost50% of the clear field value. The normalized mean square error ishowever more contained and is only about 4.9% if calculated within a 3μm by 3 μm area or 3.6% if calculated within the 0.4 μm by 0.4 μm openarea of the hole, as shown in FIG. 12. From the work presented so far,it should be obvious that neither of these error levels is immediatelyrelevant in the simulation of the image formation. What is rather morerelevant is the error incurred on the diffraction orders that arecollected by the imaging optics. If the spatial frequencies of the errordepicted in FIG. 12(c) are sufficiently high, this error will bediscarded by the projection lens. This concept is shown in FIG. 13,where the spectra (magnitude) of the scattered fields from edge-DDM andfrom rigorous 3D simulation are shown in FIG. 13(a) and (b)respectively. The marginal circles indicating the propagating planewaves (with |{right arrow over (k)}|≦2π/λ) and the collected by theimaging system plane waves (with |{right arrow over (k)}|≦2πNA(1+σ)/λR),for NA=0.75, R=4 and σ=0.3, are indicated on these plots. The magnitudeof the error in the spectrum of the field obtained from edge-DDM isshown in FIG. 13. Observe that most of the error is concentrated in thehigher spatial frequencies for |k_(x)| approaching 0.03 and only smalllevels of error exist in the collected orders. The normalized meansquare errors of all propagated and collected orders are 2.2% and 2E-4respectively.

Similarly, the amplitude of the scattered fields across the observationplane below the isolated island obtained with the edge-DDM and withfully rigorous 3D simulation and the amplitude of the error are shown inFIG. 14. Again, a locally significant error amplitude near the comers isevident. However, the energy of this error is concentrated mostly at thehigher spatial frequencies, for |k_(x)| approaching 0.03 and the errorin the collected orders is seen to be small, only 0.16% in a NMSE sense.Note that the large DC component of the isolated island mask layout wassubtracted in order to make meaningful comparisons of the edge-DDM withfully rigorous simulations. Otherwise this large DC component wouldartificially show an even better accuracy, since the error level incomparison with a much stronger signal (containing the DC term) would besmaller.

Each fully rigorous mask simulation of the patterns of FIG. 12 and FIG.14 that was performed with TEMPEST used approximately 1.8 Gbytes ofmemory and required almost 2 days on a 550 Mhz CPU. On the other hand,once all of the required 2D edge simulation results are available thescattered field can be obtained with the edge-DDM in less than 1 sec.The simulation of the isolated edge-diffractions is an off-lineprocedure. It only needs to be performed once and the edge-diffractionresults are recycled as many times as are necessary for thereconstruction of the scattered fields of arbitrary layouts. In theabove examples, it took approximately 5 min for each rigorous,isolated-edge 2D simulation.

The edge-DDM is used here to generate the scattered field from a threelevel (0°/90°/270°) alt. PSM. A portion of the arbitrary target layoutis shown in FIG. 11 before and after the phase assignment. Rectangle andedge extraction is performed within the 3 μm by 4 μm (1×) coherencewindow shown and the scattered field obtained with the edge-DDM insidethe 12 μm by 16 μm (4×) coherence window is also shown in FIG. 11 for TEand TM illuminations at λ=193 nm. Note that a fully rigorous 3D masksimulation of a 12 μm by 16 μm (4×) layout would require close to 40Gbytes of memory and would only be feasible on a multi-CPU architecture.The scattered field is obtained with the edge-DDM implementation in lessthan 1 min. This time includes the polygon and edge extraction and thefield synthesis from pre-calculated edge-diffractions. Therefore, theresults of the edge-DDM cannot be immediately compared with a fullyrigorous solution, but the accuracy of the edge-DDM within the imagingsystem's collection ability can reveal the inaccuracies of the idealk-mask, where all transitions are assumed perfectly sharp. This is donein FIG. 16, where the images (at the 30% intensity level) obtained withthe k-mask and the edge-DDM for two different undercuts (0 nm and 50 nm)are compared. The system parameters are λ=193 nm, NA=0.7, σ=0.3 and R=4.The inaccuracies of the k-mask model are revealed at various locationsin the image as seen in FIG. 16. The effect on the image of changing theundercut from 0 nm to 50 nm can be quickly and accurately evaluated.Observe the subtle differences evident in the intensity contours of the0 nm and 50 nm masks of FIG. 16.

Although the edge-DDM presented earlier achieves excellent accuracy andtremendous speed-ups as compared to a fully rigorous mask simulation,there is still one more parameter to exploit in order to furtherspeed-up the method. There is no need to carry all the details of thetrue field scattering as far as the imaging process is concerned (withlarger than 1 reduction factors), since the higher spatial frequenciesare not used. If a lumped parameter model can be devised that closelymatches the TTL spectrum of the scattered field then the task of anaccurate field representation is accomplished. A rect-function, a raisedcosine function and a Gaussian function can successfully mimic the TTLspectra of scattered fields from various size and profile openings. Asimilar principle can be applied here with the purpose of matching theTTL spectra of edgediffracted fields. Assuming a restriction topiecewise constant models, the scattered field from an isolated edgeunder TE or TM illumination can be approximated using the multistepfunctions shown in FIG. 17. Although not shown in FIG. 17, theamplitude, phase and size of all steps are optimized to achieve the bestspectral matching of the collected (TTL) orders with the ordersresulting from the continuous edge-diffracted field. Note that the keyfactor for a good spectral match is to capture the finite extent of the“off-on” edge-transition. A piecewise constant function is probably notthe best choice to achieve this task, but on the other hand a piecewiseconstant function offers a good lumped parameter approach and it lendsitself nicely as an extension of the unmatched Kirchhoff approach, whereall edge-transitions are assumed perfectly sharp. Instead ofapproximating the true edge-diffraction by a single step function(geometrical or Kirchhoff approximation) a multi-step piecewise constantfunction is employed. The amplitudes, phases and sizes of all steps areoptimized to achieve the best spectral matching of the collected orderswith the orders resulting from the continuous edge-diffracted field. Thematched bandwidth approximations to the field diffracted by a 180° edgewith 50 nm undercut under TE and TM illuminations are shown in (a) and(b) respectively.

If the matched bandwidth edge-DDM is applied on the 400 nm by 400 nm(4×) isolated hole of FIG. 12 the scattered field across the observationplane can be represented in the compact way shown in FIG. 18(a). Recallthat the edges parallel to the y-axis “see” TE polarization and theedges parallel to the x-axis TM polarization. The multi-step matched BWapproximations to the edge-diffraction shown in FIG. 17 for TE and TMilluminations intermingle in the x- and y-directions to produce themulti-color mask of FIG. 18(a), where each color represents a differentamplitude and phase region of the mask. The spectrum of the error ofthis approach is depicted in FIG. 18(b). Note that compared to the errorshown in FIG. 13(c) this error has increased by almost four-fold.Nevertheless, the error in the TTL spectrum still remains at low enoughlevels for accurate imaging simulations. A better approximatingfunction, other than the piecewise constant step function, that is morecapable of capturing the true edge-diffraction profile (such as linearor higher order polynomial) should result in better spectral matchingand consequently more accurate imaging simulations.

Consider now a domain decomposition strategy appropriate for handlingwavelength size phase defects. The bright future for alt. PSMs as amajor resolution enhancement technique (RET) has been plagued primarilyby the difficulty to reliably inspect them, flag locations that phasedefects are present and subsequently repair the defective locations.State-of-the-art inspection systems utilize a focused laser beam thatcan be either at-wavelength (up to λ=248 nm), meaning that thewavelength of the inspection beam is the same as the wavelength that thealt. PSM was designed for, or emit light of a larger wavelength. Thisbeam scans the entire mask and the reflected or transmitted scatteredfields from the particular mask location under inspection are imagedwith a very high resolution imaging system that typically has areduction factor of R=1 and NA close to 1. But here is the caveat: Afterthe signal from the inspection tool for a particular mask location isavailable it needs to be compared to something in order to determinewhether or not a phase-defect (or other defect) is present within thatlocation. Such a comparison is in general very difficult, because notonly does it need to be accurate in order to reliably flag defectivelocations, but it also needs to be rapid enough for the entire reticleto be inspected in a reasonable amount of time.

The most promising direction that has been adopted by researchersworking on the problem has been what is known as a die-to-databasecomparison. By that, it is meant that the local signal is compared tothe signal that the same mask location would produce if it weredefect-free. The problem now shifts to building the database of signalsfrom defect-free masks. Note that all possible mask geometries that willbe encountered in every inspected reticle need to exist in the database.One way to building the database would be to produce a (set of) testreticle(s) that includes every single geometry situation that isanticipated to exist in all designed layouts to come. This (set of) testreticle(s) is then meticulously inspected with a reliable tool (maybe anatomic force microscope—AFM) to guarantee that it is defect-free andfinally all signals from the different locations are collected andplaced in the vault (database). Clearly, the immensity of this task hasto do not only with the a priori precise anticipation of what layoutsituations to take provision for, not only with manufacturing the testreticle(s), characterizing it and building the database, but also withthe sheer volume of data that such a database would contain. Moreover, aprecise diagnosis of the type, size and location of a phase defectrequires an even larger database of benchmark signals, where everypossible combination of defect type, size and location in every possiblelayout configuration needs to exist in the database. Performing such atask in a way similar to the one outlined is impractical.

Simulation can again come to the rescue, at least in principle. If asimulation tool that can rapidly and accurately predict the expectedsignal resulting from a defect-free or defective location on the mask isavailable, then the task of building the database is simplifiedtremendously. Moreover, if such a tool is really fast (and accurate)then a database is not even needed. The required benchmark signal fromthe defect-free layout is generated in-situ, while the inspection systemgathers measurements. The problem with this solution is that to datethere exists no simulation tool that possesses both the rapidity andaccuracy properties. The family of simulation tools that rigorouslysolve Maxwell's equations around the reticle (such as TEMPEST) andsubsequently use a vectorial formulation for the image formation tocalculate the expected optical signal from the inspection system areimpractical because of large memory and time requirements. On the otherhand the accuracy of speedier simulation programs that circumvent thesolution of Maxwell's equations around the mask is unacceptable.

Yet another important consideration of an inspection system for alt.PSMs should be the following: Phase-defects that are not criticallyaffecting the image that a projection printing tool will produce on thewafer surface should be discarded, i.e. not flagged and not repaired.But this is also a difficult task, since the only bullet-proof way ofachieving this would be to expose wafers with the suspect defect presentand examine if the printed resist images have intolerable artifacts.Clearly, such a process now involves the close cooperation of theinspection system with the projection printing tool. However, anaccurate and properly calibrated simulation tool can alternativelyassess the tendency of the defect to print or cause other imageartifacts. Consequently, the costly step of exposing wafers can beavoided and a decision of repairing or not the defect rests fully uponthe simulation result. Again, to date, results from no simulation tool,that is fast enough in calculating expected images with phase defectspresent, can be trusted to base repair decisions on.

The domain decomposition methods described above are appropriate forsolving the problem of the rapid and accurate evaluation of thebenchmark signal from nondefective masks for the inspection system. Adomain decomposition method is presented that addresses the problem ofthe defect printability assessment, after a defect is previously found.The generation of benchmark signals from defective mask locations canalso be based on the domain decomposition technique, although more workwill be needed to adapt it and test it for the high-NA inspection opticswith R=1. As will soon be obvious, the speed and efficiency of themethod are inherited by virtue of the domain decomposition methodspresented above, although for simplicity edge-DDM is not explicitlyemployed in the following. In the simulation examples, the illuminationwavelength is 193 nm and the phase-wells are designed accordingly, so asto provide the required phase shifts for that wavelength.

Consider the example depicted in FIG. 19. In FIG. 19(a) the nearscattered field below a small layout of a 0°/180° alt. PSM is shown whena square 120 nm×120 nm (4×), 180° post phase defect is present atvarious locations within the layout. Note that although neither thelayout nor the phase defect geometry change, a different simulationaltogether is needed because of the different relative positions of thedefect within the layout. Each one of the nine 3D simulations in thematrix shown in FIG. 19(a) takes approximately 10 hrs on a 450 MHz CPUand utilize 300 Mb of memory. Subsequently, the aerial images for allnine situations are calculated for an imaging system with R=4, NA=0.75and σ=0.3. Then, the fractional change of CD_(aerial) at the middle andat the end of the line because of the defect is found for all ninedefect locations and is shown in FIG. 19(b). From the plot in FIG. 19(b)it can be deduced that when the defect is located in the middle of thephase wells or is mostly “tucked” underneath the absorbing layer(columns 3 and 1 of FIG. 19(a) respectively) the CD_(aerial) is notcritically affected, but locations such as those in column 2 of FIG.19(a) cause larger CD_(aerial) variations. In the former cases thedefects could probably be discarded whereas in the latter repair seemsnecessary. Clearly, the luxury of the 10 hr-long simulations is onlyavailable as a proof of concept and such an approach is not viable forfull-chip characterizations.

Now, consider the paradigm shown in FIG. 20. A domain decompositionmethod is again invoked, where instead of a one-step simulation of thedefective layout two separate rigorous simulations are performed, onewith just the defect-free layout and one with just the defect in clearsurroundings. The scattered field below the defect is then shiftedaccordingly, such that the defect is effectively placed at the locationit appears in the defective mask, and the uniform background light issubtracted, so as to retrieve the (complex) signal of the localperturbation that the defect causes. This perturbation is added to thedefect-free layout signal for an approximation to the signal from therigorous simulation of the defective mask. The non-defective masksimulation can be performed rapidly using the edge-DDM decompositionmethods described above.

A strategy can now be devised, where a database of the nearelectromagnetic field scattered from isolated phasedefects in clearsurroundings is created for all possible potential defects that arecritical. The volume of such a database can be reduced based on theearlier observations that defects with similar footprints (lateraldimensions) and small shape perturbations are equivalent from anelectromagnetic point-of-view.

The suggested decomposition method will be introduced through thefollowing example: Suppose a 150 nm (1×), 1:1.5 semi-dense, 90°/270°alt. PSM contact mask. The layout and the scattered field across theobservation plane under normal incidence Ey (TE) polarization are shownin FIG. 21.

Now, suppose that a 200 nm by 200 nm (4×) 90° post defect is present inthe center of the bottom left hole. According to the paradigm of FIG.20, the mask scattering simulation can be broken up into two constituentparts. This approach is shown in FIG. 22. In FIG. 22(a) the phase defectis simulated in an isolated configuration at the location that itappears in the layout. In FIG. 22(b) and (c) the near field across theobservation plane below the defective alt. PSM is shown, using thedomain decomposition method and the complete mask respectively. Thedifferences are not discernible from the plots of FIGS. 22(b) and (c)and one has to look at the amplitude of the error in FIG. 22(d). Again,not only is the error level of the decomposition low, but the frequencycontent is very high, so that it will be filtered out by the imagingsystem.

Next, suppose that the same 90° post defect is present in the center ofthe bottom right hole. The important observation here is that no newsimulation is necessary for the edge decomposition method. Thescattering of the isolated phase defect can be recycled from thedatabase, i.e., from FIG. 22(a). However, the scattered field is nowproperly shifted in the lateral x-direction and also (a key step) it ispropagated in the z-direction. This results in the plot of FIG. 23(a),where a larger diffraction spreading is evident. In FIGS. 23(b) and (c)the near field across the observation plane below the defective alt. PSMis shown, using the domain decomposition method and the complete maskrespectively. The amplitude of the error is shown in FIGS. 23(d) andsimilar observations apply in FIG. 22(d).

The near fields across cut-lines passing through the center of thebottom two holes (90°, 270°) are shown for the two defective masks inFIG. 24, comparing the rigorous (r-mask) and the decomposition methods(DDM). The non-defective case is also plotted for comparison.

Next, the images of both rigorous and DDM approaches, for both defectivemasks are compared in FIG. 25, (a) for the 90° hole and (b) for the 270°hole. The optical system parameters are λ=193 nm, R=4, NA=0.75 andσ=0.3. The agreement between the r-mask and the DDM is good, anormalized mean square error of less than 0.3% is incurred (measuredonly in the area of the defective hole). The k-mask model isinsufficient for capturing these effects and is only shown forcomparison.

The reason of the excellent agreement in the predictions of the aerialimage can be traced back to the plots of FIG. 22(d) and FIG. 23(d). Thehigh spatial frequency variation of the error means that it is mapped atthe extremities of the spectrum of propagating plane waves and is notcollected by the optical system. The normalized spectra (magnitude) ofthe near field errors are shown in FIG. 26. Observe that the impliedaccuracy from these plots should be enough even for R=1 and NA˜0.8-0.9,which would be sufficient for inspection simulations.

Thus the invention includes a suitable extension of the domaindecomposition framework for the simulation of alternating phase shiftmask with phase defects. The value of this approach is believed to besignificant in the problem of rapid assessment of defect printabilityonce a defect has been found, or in the die-to-database comparisons ofinspection systems when deciding about the existence or not of aphase-defect.

The invention is an extension of known domain decomposition methods thatintroduces tremendous versatility, since only a small number of isolatededge-diffraction simulations is shown to contain all the necessaryinformation for the synthesis of the scattered field from arbitrary 2Dmask layouts and subsequent accurate imaging simulations. The limits ofthis method are reached when the mask features are smaller than awavelength in size and the vertical mask topography is large. Through asystematic process it was determined that features as small as at least200 nm bearing 270° of phase-wells can be accurately decomposed via theedge-DDM, at λ=193 nm. The method was tested for simple 2D layouts,where rigorous mask simulations are possible. Excellent accuracyaccompanied by speed-up factors of 172,800 (1 sec vs. 2 days) have beendemonstrated. The accuracy of the method is attributed to the fact thaton one hand the edge-diffraction phenomena are modeled rigorouslythrough 2D edge-diffraction simulations and that on the other hand theerrors incurred during the synthesis (primarily near the comers) aremapped at the extremities of the spectrum of propagating plane waves anddo not contribute to the image formation. It was emphasized that allnecessary rigorous 2D simulations of the diffraction from isolated edgeprofiles is performed off-line and recycled for the diffractioncalculation of arbitrary layouts. If needed, the accuracy of the edgeprofile diffraction simulations can be pushed to extreme limits, sinceit is performed only once. This can aid for example in the correctsimulation of unusual edge profiles whose geometrical details wouldrequire excessive discretization of the domain. It was also shown thatthe exact details of the edge-diffraction are not necessary for accurateimaging simulations (with larger than 1 reduction factors) and apiecewise constant, multistep edge-transition model that matches the TTLspectrum was introduced. The range of validity of the edge-DDM isexpected to be appropriate for rapid and accurate evaluation of aerialimages whenever speed is critical, as for example in full-chip OPCsoftware and die-to-database comparisons in the inspection of alt. PSMs.

While the invention has been described with reference to specificembodiments, the description is illustrative of the invention and is notto be construed as limiting the invention. Various modifications andapplications may occur to those skilled in the art without departingfrom the true scope and spirit of the invention as defined by theappended claims.

What is claimed is:
 1. A method for simulation of light scattering inlarge three dimensional objects in a optical imaging system comprisingthe steps of: a) decomposing a three dimensional object into a pluralityof elementary two dimensional objects, b) defining the two dimensionalobjects by edges of the objects, c) applying edge domain decompositionfor the edges of the to two dimensional objects corresponding to a threedimensional object to identify light scattering by the associated onedimensional edges, and d) summing scattered light from all edges tosimulate light scattering in two three dimensions.
 2. The method ofclaim 1 and further including: e) spectrally mapping the edge domaindecomposition.
 3. The method of claim 2 wherein the three dimensionalobject comprises a photo mask with multiple and interconnected openings.4. The method of claim 3 wherein step c) applying edge domaindecomposition includes computational methods which speed-up calculationsthrough use of table look-up of previously calculated objects andfiltering that sequentially invokes more rigorous simulation as needed.5. The method of claim 4 wherein an electromagnetic field from thediffraction of isolated edges is recycled in the synthesis of a neardiffracted field for arbitrary two dimensional diffracting geometries.6. The method of claim 5 wherein the a scattered field from all edgesare summed to simulate light scattering in by the two three dimensionalobjects object.
 7. The method of claim 6 wherein scattered light ispolarized and step c) applying edge domain decomposition includesseparate decomposition for light parallel to edges (TE) and for lightperpendicular (TM) to edges.
 8. The method of claim 7 wherein step c)applying edge domain decomposition includes pre-simulating all distinctedges in the object with all possible illumination directions and fieldpolarizations.
 9. The method of claim 8 further comprising decomposingthe three dimensional object into a plurality of the elementary twodimensional objects, wherein in step a) the two dimensional objectscomprise rectangles.
 10. The method of claim 9 wherein the object is aphase shifting mask and step c) of pre-simulating light scattering fromedges includes phase effects of etch depths in defining the phaseshifting mask.
 11. The method of claim 10 wherein the simulation oflight scattering is based on through the a lens optical spectra.
 12. Themethod of claim 10 wherein the simulation of light scattering is basedon a raised cosine function for the optical spectra.
 13. The method ofclaim 10 wherein the simulation of light scattering is based on aGaussian function for the optical spectra.
 14. The method of claim 10wherein the simulation of light scattering is based on a rect-functionfor the optical spectra.
 15. The method of claim 3 wherein the object isa phase shifting mask and step c) applying edge domain decompositionincludes pre-simulating light scattering from edges and includes phaseeffects of etch depths in defining the phase shifting mask.
 16. Themethod of claim 15 wherein scattered light is polarized and step c)includes separate decomposition for light parallel to edges (TB) (TE)and for light perpendicular (TM) to edges.
 17. The method of claim 16wherein step c) applying edge domain decomposition includespre-simulating all distinct edges in the object with all possibleillumination directions and field polarizations.
 18. The method of claim1 wherein step c) applying edge domain decomposition includespre-simulating all distinct edges in the object with all possibleillumination directions and field polarizations.
 19. A methodcomprising: applying edge-domain decomposition to the a plurality ofone-dimensional edges derived from a three-dimensional object toidentify light scattering by the one-dimensional edges; and using acomputer, summing scattered light from the one-dimensional edges tosimulate the light scattering by the three-dimensional object.
 20. Themethod of claim 19, further comprising spectrally mapping the edgedomain decomposition.
 21. The method of claim 19, wherein the threedimensional object comprises a photo mask with multiple andinterconnected openings.
 22. The method of claim 19, wherein theapplying edge-domain decomposition comprises using computational methodsthat speed up calculations by using lookup tables of previouslycalculated objects and filtering that sequentially invokes more rigoroussimulation as needed.
 23. The method of claim 19, wherein anelectromagnetic field from diffraction of isolated edges is recycled insynthesis of a near diffracted field for arbitrary two-dimensionaldiffracting geometries.
 24. The method of claim 19, wherein one or morescattered fields from the one-dimensional edges are summed to simulatethe light scattering by the three-dimensional object.
 25. The method ofclaim 19, wherein the scattered light is polarized and the applyingedge-domain decomposition includes performing separate decompositionsfor light parallel to the one-dimensional edges (TE) and for lightperpendicular (TM) to the one-dimensional edges.
 26. The method of claim19, wherein the applying edge-domain decomposition includes distinctedges in the three-dimensional object with one or more possibleillumination directions and field polarizations.
 27. The method of claim26, wherein the three-dimensional object is a phase-shifting mask andthe pre-simulating accounts for phase effects of etch depths in thephase shifting mask.
 28. The method of claim 19, wherein two-dimensionalobjects defining the three-dimensional object comprise rectangles. 29.The method of claim 19, wherein the applying edge-domain decompositionis based on through-the-lens optical spectra.
 30. The method of claim19, wherein the applying edge-domain decomposition is based on a raisedcosine function for optical spectra.
 31. The method of claim 19, whereinthe applying edge-domain decomposition is based on a Gaussian functionfor optical spectra.
 32. The method of claim 19, wherein the applyingedge-domain decomposition is based on a rect-function for opticalspectra.
 33. The method of claim 19, wherein: the three-dimensionalobject is a phase shifting mask, and the applying edge-domaindecomposition includes pre-simulating light scattering from theone-dimensional edges and phase effects of etch depths in the phaseshifting mask.
 34. The method of claim 33, wherein: the scattered lightis polarized; and the applying edge-domain decomposition includesperforming separate decompositions for light parallel to theone-dimensional edges (TE) and for light perpendicular (TM) to theone-dimensional edges.
 35. The method of claim 34, wherein the applyingedge-domain decomposition includes pre-simulating distinct edges in thethree-dimensional object with a plurality of possible illuminationdirections and field polarizations.
 36. The method of claim 19, whereinthe applying edge-domain decomposition includes pre-simulating distinctone-dimensional edges in the three-dimensional object with a pluralityof pre-selected possible illumination directions and fieldpolarizations.
 37. A non-transitory computer-readable storage mediumstoring computer-executable instructions for causing a computer toperform a method comprising: applying edge-domain decomposition to aplurality of one-dimensional edges composing a three-dimensional objectto identify light scattering by the one-dimensional edges; and summingscattered light from the one-dimensional edges to simulate the lightscattering by the three-dimensional object.
 38. The non-transitorycomputer-readable storage medium of claim 37, wherein the method furthercomprises spectrally mapping the edge domain decomposition.
 39. Thenon-transitory computer-readable storage medium of claim 37, wherein thethree dimensional object comprises a photo mask with multiple andinterconnected openings.
 40. The non-transitory computer-readablestorage medium of claim 37, wherein the applying edge-domaindecomposition comprises using computational methods that speed upcalculations by using look-up tables of previously calculated objectsand filtering that sequentially invokes more rigorous simulation asneeded.
 41. The non-transitory computer-readable storage medium of claim37, wherein an electromagnetic field from diffraction of isolated edgesis recycled in synthesis of a near diffracted field for arbitrarytwo-dimensional diffracting geometries.
 42. The non-transitorycomputer-readable storage medium of claim 37, wherein one or morescattered fields from the one-dimensional edges are summed to simulatethe light scattering by the three-dimensional object.
 43. Thenon-transitory computer-readable storage medium of claim 37, wherein thescattered light is polarized and the applying edge-domain decompositionincludes performing separate decompositions for light parallel to theone-dimensional edges (TE) and for light perpendicular (TM) to theone-dimensional edges.
 44. The non-transitory computer-readable storagemedium of claim 37, wherein the applying edge-domain decompositionincludes pre-simulating distinct edges in the three-dimensional objectwith one or more possible illumination directions and fieldpolarizations.
 45. The non-transitory computer-readable storage mediumof claim 37, wherein two-dimensional objects defining thethree-dimensional object comprise rectangles.
 46. The non-transitorycomputer-readable storage medium of claim 44, wherein thethree-dimensional object is a phase-shifting mask and the pre-simulatingaccounts for phase effects of etch depths in the phase shifting mask.47. The non-transitory computer-readable storage medium of claim 37,wherein the applying edge-domain decomposition is based onthrough-the-lens optical spectra.
 48. The non-transitorycomputer-readable storage medium of claim 37, wherein the applyingedge-domain decomposition is based on a raised cosine function foroptical spectra.
 49. The non-transitory computer-readable storage mediumof claim 37, wherein the applying edge-domain decomposition is based ona Gaussian function for optical spectra.
 50. The non-transitorycomputer-readable storage medium of claim 37, wherein the applyingedge-domain decomposition is based on a rect-function for opticalspectra.
 51. The non-transitory computer-readable storage medium ofclaim 37, wherein: the three-dimensional object is a phase shiftingmask, and the applying edge-domain decomposition includes pre-simulatinglight scattering from the one-dimensional edges and phase effects ofetch depths in the phase shifting mask.
 52. The non-transitorycomputer-readable storage medium of claim 51, wherein: the scatteredlight is polarized, and the applying edge-domain decomposition includesperforming separate decompositions for light parallel to theone-dimensional edges (TE) and for light perpendicular (TM) to theone-dimensional edges.
 53. The non-transitory computer-readable storagemedium of claim 52, wherein the applying edge-domain decompositionincludes pre-simulating distinct edges in the three-dimensional objectwith a plurality of possible illumination directions and fieldpolarizations.
 54. The non-transitory computer-readable storage mediumof claim 37, wherein the applying edge-domain decomposition includespre-simulating distinct one-dimensional edges in the three-dimensionalobject with a plurality of pre-selected possible illumination directionsand field polarizations.
 55. A method of simulating an aerial imageprojected by a multiple-opening mask having a plane defined by an X-axisand a Y-axis, comprising: receiving layout data representing physicaldimensions of one or more openings in the mask; producing firstscattered field data for a selected one of the openings using a firsttwo-dimensional model oriented along the X-axis; producing secondscattered field data for the selected opening using a secondtwo-dimensional model oriented along the Y-axis; producing thirdscattered field data for the selected opening using a Kirchoff maskmodel; and using a computer, producing composite scattered field datafor the selected opening from the first scattered field data, the secondscattered field data, and the third scattered field data.
 56. The methodof claim 55, wherein the layout data includes data concerning phaseshift properties of the one or more openings.
 57. The method of claim55, wherein the layout data includes data concerning binary maskproperties of the one or more openings.
 58. The method of claim 55,wherein the X-axis is oriented in a TM direction for a polarizedelectromagnetic wave to be transmitted through the selected opening andthe Y-axis is oriented in a TE direction for the polarizedelectromagnetic wave.
 59. The method of claim 55, wherein the methodfurther comprises decomposing the data representing the physicaldimensions of the one or more openings into data defining the one ormore openings as rectangles.
 60. The method of claim 59, wherein thedecomposing produces mutually disjoint rectangles.
 61. The method ofclaim 55, further comprising performing optical proximity correctionusing the composite scattered field data.
 62. The method of claim 55,further comprising performing die-to-database inspection using thecomposite scattered field data.
 63. The method of claim 55, wherein theproducing the composite scattered field data is performed by linearlycombining the first scattered field data, the second scattered fielddata, and the third scattered field data.
 64. The method of claim 55,wherein the first scattered field data and the second scattered fielddata are produced using pre-calculated values stored in one or morelook-up tables.
 65. A non-transitory computer-readable storage mediumstoring computer-executable instructions for causing a computer toperform a method of simulating an aerial image projected by amultiple-opening mask having a plane defined by an X-axis and a Y-axis,the method comprising: receiving layout data representing physicaldimensions of one or more openings in the mask; producing firstscattered field data for a selected one of the openings using a firsttwo-dimensional model oriented along the X-axis; producing secondscattered field data for the selected opening using a secondtwo-dimensional model oriented along the Y-axis; producing thirdscattered field data for the selected opening using a Kirchoff maskmodel; and producing composite scattered field data for the selectedopening from the first scattered field data, the second scattered fielddata, and the third scattered field data.
 66. The non-transitorycomputer-readable storage medium of claim 65, wherein the layout dataincludes data concerning phase shift properties of the one or moreopenings.
 67. The non-transitory computer-readable storage medium ofclaim 65, wherein the layout data includes data concerning binary maskproperties of the one or more openings.
 68. The non-transitorycomputer-readable storage medium of claim 65, wherein the X-axis isoriented in a TM direction for a polarized electromagnetic wave to betransmitted through the selected opening and the Y-axis is oriented in aTE direction for the polarized electromagnetic wave.
 69. Thenon-transitory computer-readable storage medium of claim 65, wherein themethod further comprises decomposing the data representing the physicaldimensions of the one or more openings into data defining the one ormore openings as rectangles.
 70. The non-transitory computer-readablestorage medium of claim 69, wherein the decomposing produces mutuallydisjoint rectangles.
 71. The non-transitory computer-readable storagemedium of claim 65, wherein the method further comprises performingoptical proximity correction using the composite scattered field data.72. The non-transitory computer-readable storage medium of claim 65,wherein the method further comprises performing die-to-databaseinspection using the composite scattered field data.
 73. Thenon-transitory computer-readable storage medium of claim 65, wherein theproducing the composite scattered field data is performed by linearlycombining the first scattered field data, the second scattered fielddata, and the third scattered field data.
 74. The non-transitorycomputer-readable storage medium of claim 65, wherein the firstscattered field data and the second scattered field data are producedusing pre-calculated values stored in one or more look-up tables.
 75. Amethod of performing optical proximity correction for a mask,comprising: receiving layout data representing the mask, the layout datadefining polygons and phase assignments; assigning edge types to edgesof one or more of the polygons, wherein the edge type for a respectiveedge is based at least in part on phase assignment data for an areaadjacent to the respective edge; simulating an aerial image produced bythe mask using two-dimensional models for the edges of the one or morepolygons, wherein the two-dimensional models are selected based at leastin part on the respective edge types of the edges and wherein thesimulating produces edge image intensities; and using a computer,modifying the layout data by applying an optical proximity correctionmethod to the layout data using the aerial image.
 76. The method ofclaim 75, wherein the edge type is one of a Cr-layer/0° edge, aCr-layer/90° edge, a Cr-layer/270° edge, a 0°/90° edge, or a 0°/270°edge.
 77. The method of claim 75, further comprising decomposing one ormore of the polygons in the layout data into mutually disjointrectangles.
 78. The method of claim 75, wherein the polygons comprisenon-Manhattan geometries.
 79. The method of claim 75, wherein the aerialimage is simulated as being produced by quadrupole illumination.
 80. Themethod of claim 75, wherein the modifying includes adding one or more ofan edge shift or a sub-resolution assist feature to the layout data. 81.The method of claim 75, wherein the simulating ignores corner effects.82. The method of claim 75, wherein the simulating includes using one ormore look-up tables to determine the aerial image.
 83. The method ofclaim 82, wherein the one or more look-up tables are generated bycalculating electromagnetic field solutions.
 84. A non-transitorycomputer-readable storage medium storing computer-executableinstructions for causing a computer to perform an optical proximitycorrection method for a mask, the method comprising: receiving layoutdata representing the mask, the layout data defining polygons and phaseassignments; assigning edge types to edges of one or more of thepolygons, wherein the edge type for a respective edge is based at leastin part on phase assignment data for an area adjacent to the respectiveedge; simulating an aerial image produced by the mask usingtwo-dimensional models for the edges of the one or more polygons,wherein the two-dimensional models are selected based at least in parton the respective edge types of the edges and wherein the simulatingproduces edge image intensities; and modifying the layout data byapplying an optical proximity correction method to the layout data usingthe aerial image.
 85. The non-transitory computer-readable storagemedium of claim 84, wherein the edge type is one of a Cr-layer/0° edge,a Cr-layer/90° edge, a Cr-layer/270° edge, a 0°/90° edge, or a 0°/270°edge.
 86. The non-transitory computer-readable storage medium of claim84, wherein the method further comprises decomposing one or more of thepolygons in the layout data into mutually disjoint rectangles.
 87. Thenon-transitory computer-readable storage medium of claim 84, wherein thepolygons comprise non-Manhattan geometries.
 88. The non-transitorycomputer-readable storage medium of claim 84, wherein the aerial imageis simulated as being produced by quadrupole illumination.
 89. Thenon-transitory computer-readable storage medium of claim 84, wherein themodifying includes adding one or more of an edge shift or asub-resolution assist feature to the layout data.
 90. The non-transitorycomputer-readable storage medium of claim 84, wherein the simulatingignores corner effects.
 91. The non-transitory computer-readable storagemedium of claim 84, wherein the simulating includes using one or morelook-up tables to determine the aerial image.
 92. The non-transitorycomputer-readable storage medium of claim 91, wherein the one or morelook-up tables are generated by calculating electromagnetic field.
 93. Amethod of inspecting a mask, comprising: receiving data produced byscanning a beam across a surface of the mask and imaging scatteredelectromagnetic fields that result from the scanning; and comparing thereceived data to benchmark data, wherein the benchmark data is producedby applying an edge domain decomposition method (EDDM) to layout datarepresenting the mask.
 94. The method of claim 93, wherein the benchmarkdata is produced at substantially a same time as the scanning.
 95. Themethod of claim 93, further comprising generating a database storing thebenchmark data.
 96. The method of claim 93, wherein the benchmark datarepresents a defect-free version of the mask.
 97. The method of claim93, further comprising outputting a signal indicating whether the maskis defective based on the comparing.
 98. The method of claim 93, whereinthe benchmark data includes data produced by: defining a defect-freearea of the mask; simulating the defect-free area of the mask using theEDDM; and simulating a defect within the defect-free area of the mask.99. The method of claim 98, wherein the simulating a defect includesevaluating an aerial critical dimension (CD).
 100. A non-transitorycomputer-readable storage medium storing computer-executableinstructions for causing a computer to perform a method, the methodcomprising comparing received data to benchmark data, wherein thereceived data is produced at least in part by scanning a beam across asurface of a mask to produce a scattered electromagnetic field, andwherein the benchmark data is produced at least in part by applying anedge domain decomposition method (EDDM) to layout data representing themask.
 101. The non-transitory computer-readable storage medium of claim100, wherein the benchmark data is produced at substantially a same timeas the scanning.
 102. The non-transitory computer-readable storagemedium of claim 100, wherein the benchmark data is stored in a database.103. The non-transitory computer-readable storage medium of claim 100,wherein the benchmark data represents a defect-free version of the mask.104. The non-transitory computer-readable storage medium of claim 100,wherein the method further comprises, based on the comparing, outputtinga signal indicating whether the mask is defective.
 105. Thenon-transitory computer-readable storage medium of claim 100, whereinthe benchmark data includes data produced by: defining a defect-freearea of the mask; simulating the defect-free area of the mask using theEDDM; and simulating a defect within the defect-free area of the mask.106. The non-transitory computer-readable storage medium of claim 105,wherein the simulating a defect includes evaluating an aerial criticaldimension (CD).
 107. A method for simulating light scattering by athree-dimensional object in an optical imaging system, the methodcomprising: applying edge-domain decomposition to the three-dimensionalobject to identify light scattering by one-dimensional edges that definethe three-dimensional object; and using a computer, summing scatteredlight from all the edges to simulate light scattering in two dimensions.